This invention relates broadly to the extraction of the slowness dispersion characteristics of multiple possibly interfering signals in broadband acoustic waves, and more particularly to the processing of acoustic waveforms where there is dispersion, i.e. a dependence of wave speed and attenuation with frequency.
Dispersive processing of borehole acoustic data has been a key ingredient for the characterization and estimation of rock properties using borehole acoustic modes. The most common parameters describing the dispersion characteristics are the wavenumber, k(f), and the attenuation, A(f), both functions of the frequency f and are of great interest in characterization of rock and fluid properties around the borehole. In addition, they are of interest in a variety of other applications such as non-destructive evaluation of materials using ultrasonic waves or for handling ground roll in surface seismic applications, where the received data exhibits dispersion that needs to be estimated. The dispersion characteristic consisting of the phase and group slowness (reciprocal of velocity) are linked to the wavenumber k as follows:
                                                      o                    ⁢                      (            f            )                          =                              1                                          V                phase                            ⁡                              (                f                )                                              =                                                                                          k                    ⁡                                          (                      f                      )                                                        f                                ·                                                    g                                            ⁢                              (                f                )                                      =                                          1                                                      V                    group                                    ⁡                                      (                    f                    )                                                              =                                                ⅆ                                      k                    ⁡                                          (                      f                      )                                                                                        ⅆ                  f                                                                                        (        1        )                            where the group slowness sg and phase slowness sφ are not independent; in fact, they satisfy        
      s    g    =            s      ϕ        +          f      ⁢                        ⅆ                      s            ϕ                                    ⅆ          f                                    There is therefore the need to extract these quantities of interest from the measured acoustic data.        
One class of extraction methods uses physical models relating the rock properties around the borehole to the predicted dispersion curves. Waveform data collected by an array of sensors is back propagated according to each of these modeled dispersion curves and the model is adjusted until there is good semblance (defined below) among these back propagated waveforms indicating a good fit of the model to the data. An example is the commercial DSTC algorithm used for extracting rock shear velocity from the dipole flexural mode (see C. Kimball, “Shear slowness measurement by dispersive processing of borehole flexural mode,” Geophysics, vol. 63, no. 2, pp. 337-344, March 1998) (in this case we directly invert for the rock property of interest). One drawback of this method is that models are available for only the simpler cases and, moreover, some other physical parameters need to be known in order to carry out this process. In the case of model mismatch or error in parameter input, incorrect (biased) answers may be produced. In addition, this processing assumes the presence of only a single modeled propagating mode and pre-processing steps such as time windowing and filtering may be needed to isolate the mode of interest. These also require user input and in some cases the answers may be sensitive to the latter, requiring expert users for correctly processing the data. Another approach based on multiwave processing was proposed in C. Kimball, “Sonic well logging with multiwave processing utilizing a reduced propagation matrix,” US Patent Application Publication 2000552075A, but it too is dependent on the use of models and is applicable to a limited number of cases.
One way of addressing such drawbacks is to directly estimate the dispersion characteristics from the data without reference to particular physical models. Not only can these be used for quantitative inversion of parameters of interest but the dispersion curves carry important information about the acoustic state of the rock and are important tools for interpretation and validation. Moreover, in order to be part of a commercial processing chain, these have to be free of user inputs so as to operate in an automated unsupervised fashion. A classical method of extracting this information from data collected by an array of sensors, for example, in seismic applications, is to use a 2D FFT, also called the f-k (frequency wavenumber) transform. This indicates the dispersion characteristics of propagating waves both dispersive and non-dispersive, but is effective only with large arrays of tens of sensors. For applications with fewer sensors as commonly found in sensor arrays (2-13 sensors) on borehole wireline and LWD tools, this method does not have the necessary resolution and accuracy to produce useful answers.
A high resolution method appropriate for shorter arrays was developed using narrow band array processing techniques applied to frequency domain data obtained by performing an FFT on the array waveform data (see S. Lang, A. Kurkjian, J. McClellan, C. Morris, and T. Parks, “Estimating slowness dispersion from arrays of sonic logging waveforms,” Geophysics, vol. 52, no. 4, pp. 530-544, April 1987 and M. P. Ekstrom, “Dispersion estimation from borehole acoustic arrays using a modified matrix pencil algorithm,” ser. 29th Asilomar Conference on Signals, Systems and Computers, vol. 2, 1995, pp. 449-453). While this is an effective tool for studying dispersion behavior in a supervised setting, it produces unlabelled dots on the f-k plane and therefore is not suitable for deployment for automated unsupervised processing. Moreover the operation of FFT washes out the information pertaining to the time of arrival of various propagating modes, thereby reducing its performance as well as the effectiveness of interpretation—especially for weaker modes of interest overlapping with stronger ones in the frequency domain (see B. Sinha and S. Zeroug, “Geophysical prospecting using sonics and ultrasonics,” Schlumberger-Doll Research, Tech. Rep., August 1998). It also does not exploit the continuity across frequency of the dispersion characteristics.
Recently developed algorithms (see W. H. Prossner and M. D. Seale, “Time frequency analysis of the dispersion of lamb modes,” Journal of Acoustical Society of America, vol. 105, no. 5, pp. 2669-2676, May 1999 or A. Roueff, J. I. Mars, J. Chanussot, and H. Pedersen, “Dispersion estimation from linear array data in the time-frequency plane,” IEEE Transactions on Signal Processing, vol. 53, no. 10, pp. 3738-378, October 2005. Or S. Aeron, S. Bose, and H. P. Valero, “Automatic dispersion extraction using continuous wavelet transform,” in International Conference of Acoustics, Speech and Signal Processing, 2008 or S. Bose, S. Aeron and H. P. Valero, “Dispersion extraction for acoustic data using time frequency analysis,” 60.1752/1753, US NP, filed 2007) have been proposed using a time frequency analysis with continuous wavelet transform or short time Fourier transform along with beamforming methods. However, these require that either there is only one component or, if more than one, that they do not overlap in the time-frequency plane.